Superposition operators and functions of bounded p-variation II Original Research Article

We continue the study of the superposition operator Tf:g↦f∘gTf:g↦f∘g, on the space View the MathML sourceBVp1(I) of primitives of real-valued functions of bounded p-variation on an interval I. We give first a characterization of the functions f such that TfTf takes View the MathML sourceBVp1(I) to itself. Then we characterize the functions f for which TfTf is continuous, uniformly continuous, and differentiable, as a mapping of View the MathML sourceBVp1(I) to itself, respectively. By exploiting the Peetreʹs Imbedding Theorem and the Fubini property, we derive partial results on continuity of TfTf in Besov spaces View the MathML sourceBp,qs(Rn), for a smoothness parameter s satisfying 0